“Studies on curve singularities”

Wednesday, March 6, 2013 3:40–5 p.m.

The goal of the talk is to relate the singularity types of a rational plane curve to the syzygies of the forms parametrizing it. This is a report on joint work with Cox, Kustin, and Ulrich. More specifically, let C be a rational plane curve of degree d parametrized by three forms, which can be assumed to be of degree d as well. The syzygy matrix of this parametrization is a 2 by 3 matrix whose entries are forms of degrees d_1 and d_2, where d_1 d_2=d. Among other things we consider curves of even degree d=2c; we show that if C has a singular point (including an infinitely near singular point) of multiplicity at least c, then the multiplicity of this singularity is exactly c and furthermore d_1 = d_2 =c. We establish, essentially, a correspondence between the constellation of multiplicity c singularities on or infinitely near C on the one hand and the shapes of the syzygy matrices on the other hand. Using this, we give a stratification of the space of rational plane curves into irreducible locally closed sets, according to the constellation of singularities of maximal multiplicity c.